The Temperley–Lieb tower and the Weyl algebra

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2025-05-25 DOI:10.1112/jlms.70174

Matthew Harper, Peter Samuelson

{"title":"The Temperley–Lieb tower and the Weyl algebra","authors":"Matthew Harper, Peter Samuelson","doi":"10.1112/jlms.70174","DOIUrl":null,"url":null,"abstract":"We define a monoidal category <math>\n <semantics>\n <mi>W</mi>\n <annotation>${\\mathbf {W}}$</annotation>\n </semantics></math> and a closely related 2-category <math>\n <semantics>\n <mrow>\n <mn>2</mn>\n <mi>Weyl</mi>\n </mrow>\n <annotation>${\\mathbf {2Weyl}}$</annotation>\n </semantics></math> using diagrammatic methods. We show that <math>\n <semantics>\n <mrow>\n <mn>2</mn>\n <mi>Weyl</mi>\n </mrow>\n <annotation>${\\mathbf {2Weyl}}$</annotation>\n </semantics></math> acts on the category <math>\n <semantics>\n <mrow>\n <mi>TL</mi>\n <mo>:</mo>\n <mo>=</mo>\n <msub>\n <mo>⨁</mo>\n <mi>n</mi>\n </msub>\n <msub>\n <mo>TL</mo>\n <mi>n</mi>\n </msub>\n <mo>−</mo>\n <mi>mod</mi>\n </mrow>\n <annotation>$\\mathbf {TL}:=\\bigoplus _n \\operatorname{TL}_n\\mathrm{-mod}$</annotation>\n </semantics></math> of modules over Temperley–Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of <math>\n <semantics>\n <mi>W</mi>\n <annotation>${\\mathbf {W}}$</annotation>\n </semantics></math> and a third category we define <math>\n <semantics>\n <msup>\n <mi>W</mi>\n <mi>∞</mi>\n </msup>\n <annotation>${\\mathbf {W}}^\\infty$</annotation>\n </semantics></math> are closely related to the Weyl algebra. We formulate a sense in which <math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n <mn>0</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>W</mi>\n <mi>∞</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$K_0({\\mathbf {W}}^\\infty)$</annotation>\n </semantics></math> acts asymptotically on <math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n <mn>0</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>TL</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$K_0(\\mathbf {TL})$</annotation>\n </semantics></math>.","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 5","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70174","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70174","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We define a monoidal category $W$ and a closely related 2-category $2 Weyl$ using diagrammatic methods. We show that $2 Weyl$ acts on the category $TL : = ⨁_{n} {TL}_{n} - \mod$ of modules over Temperley–Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of $W$ and a third category we define $W^{\infty}$ are closely related to the Weyl algebra. We formulate a sense in which $K_{0} (W^{\infty})$ acts asymptotically on $K_{0} (TL)$ .

查看原文本刊更多论文

坦波利-利布塔和魏尔代数

我们用图解的方法定义了一个单类W ${\mathbf {W}}$和一个密切相关的2类2 Weyl ${\mathbf {2Weyl}}$。我们证明2 Weyl ${\mathbf {2Weyl}}$作用于类别TL：templeley - lieb代数上的模的= n TL n - mod $\mathbf {TL}:=\bigoplus _n \operatorname{TL}_n\mathrm{-mod}$，其生成的1-态由归纳和限制作用。W ${\mathbf {W}}$的Grothendieck群和我们定义的第三类W∞${\mathbf {W}}^\infty$与Weyl代数密切相关。我们给出了K 0 (W∞)$K_0({\mathbf {W}}^\infty)$作用于渐近的意义k0 (TL) $K_0(\mathbf {TL})$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.